LinearModel_Project
Parsa
2019-03-30
test <- read.csv("Admission_Predict_Ver1.1.csv")
summary (test)## Serial.No. GRE.Score TOEFL.Score University.Rating
## Min. : 1.0 Min. :290.0 Min. : 92.0 Min. :1.000
## 1st Qu.:125.8 1st Qu.:308.0 1st Qu.:103.0 1st Qu.:2.000
## Median :250.5 Median :317.0 Median :107.0 Median :3.000
## Mean :250.5 Mean :316.5 Mean :107.2 Mean :3.114
## 3rd Qu.:375.2 3rd Qu.:325.0 3rd Qu.:112.0 3rd Qu.:4.000
## Max. :500.0 Max. :340.0 Max. :120.0 Max. :5.000
## SOP LOR CGPA Research
## Min. :1.000 Min. :1.000 Min. :6.800 Min. :0.00
## 1st Qu.:2.500 1st Qu.:3.000 1st Qu.:8.127 1st Qu.:0.00
## Median :3.500 Median :3.500 Median :8.560 Median :1.00
## Mean :3.374 Mean :3.484 Mean :8.576 Mean :0.56
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:9.040 3rd Qu.:1.00
## Max. :5.000 Max. :5.000 Max. :9.920 Max. :1.00
## Chance.of.Admit
## Min. :0.3400
## 1st Qu.:0.6300
## Median :0.7200
## Mean :0.7217
## 3rd Qu.:0.8200
## Max. :0.9700head(test)## Serial.No. GRE.Score TOEFL.Score University.Rating SOP LOR CGPA Research
## 1 1 337 118 4 4.5 4.5 9.65 1
## 2 2 324 107 4 4.0 4.5 8.87 1
## 3 3 316 104 3 3.0 3.5 8.00 1
## 4 4 322 110 3 3.5 2.5 8.67 1
## 5 5 314 103 2 2.0 3.0 8.21 0
## 6 6 330 115 5 4.5 3.0 9.34 1
## Chance.of.Admit
## 1 0.92
## 2 0.76
## 3 0.72
## 4 0.80
## 5 0.65
## 6 0.90attach(test)
#Linear Regression and some plots
#Here's a linear model (Chance of Admit)
linear <- lm(Chance.of.Admit ~., data=test)
#summary(linear)
plot(linear)
#Here's a linear model (University Rating)
linear <- lm(University.Rating ~., data=test)
#summary(linear)
plot(linear)
logmod <- glm(Research~., data=test)
#summary(logmod)
plot(logmod)
chance.vs.CGPA <- lm(test$Chance.of.Admit ~ test$CGPA)
plot(test$Chance.of.Admit ~ test$CGPA, xlab = "Chance of Admission", ylab = "CGPA", main = "Chance of Admission VS CGPA")
abline(chance.vs.CGPA , col="red", lwd=3, data = test)## Warning in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): "data"
## is not a graphical parameter
#sum ((predict(chance.vs.CGPA, data.frame(test))) - test$Chance.of.Admit)^2 /nrow(test)Variable Selection for Chance of Admittion
By performing backwards selection, we will remove the least significant values until all values are significant.
linear <- lm(Chance.of.Admit~ ., data = test )
#summary(linear)
#Remove University Ranking because it has the highest non significant p value
linear <- lm(Chance.of.Admit~ GRE.Score + TOEFL.Score + SOP +LOR + CGPA + Research , data = test )
#summary(linear)
#Remove SOP has the second highest non significant p value
linear <- lm(Chance.of.Admit~ GRE.Score + TOEFL.Score +LOR + CGPA + Research , data = test )
#Remove Serial No.
linear <- lm(Chance.of.Admit~ GRE.Score + TOEFL.Score +LOR + CGPA + Research , data = test )
#summary(linear)
#plot(test)
#linearPlot <- plot(Chance.of.Admit~.,data=test)Variable Selection for Research
linear <- lm(Research~ Serial.No. + GRE.Score + TOEFL.Score + University.Rating + SOP +LOR + CGPA, data = test )
#summary(linear)
#Remove SOP
linear <- lm(Research~ Serial.No. + GRE.Score + TOEFL.Score + University.Rating +LOR + CGPA, data = test )
#summary(linear)
#Remove SOP, CGPA
linear <- lm(Research~ Serial.No. + GRE.Score + TOEFL.Score + University.Rating +LOR, data = test )
#summary(linear)
#Remove SOP, CGPA, LOR
linear <- lm(Research~ Serial.No. + GRE.Score + TOEFL.Score + University.Rating, data = test )
#summary(linear)
#Remove SOP, CGPA, LOR, TOEFL
linear <- lm(Research~ Serial.No. + GRE.Score + University.Rating, data = test )
#summary(linear)
#Remove SOP, CGPA, LOR, TOEFL, Serial Number
linear <- lm(Research~ + GRE.Score + University.Rating, data = test )
summary(linear)##
## Call:
## lm(formula = Research ~ +GRE.Score + University.Rating, data = test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.14033 -0.35017 0.00906 0.29255 1.00181
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.415603 0.625451 -10.258 <2e-16 ***
## GRE.Score 0.021546 0.002099 10.266 <2e-16 ***
## University.Rating 0.050337 0.020731 2.428 0.0155 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4089 on 497 degrees of freedom
## Multiple R-squared: 0.3254, Adjusted R-squared: 0.3227
## F-statistic: 119.9 on 2 and 497 DF, p-value: < 2.2e-16plot(linear)
##based on the Normal Q-Q Plot, we can determine that the the data fits well
Variable Selection for University Ranking
linear <- lm(University.Rating~ Serial.No. + GRE.Score + TOEFL.Score + SOP +LOR + CGPA + Research, data = test )
summary(linear)##
## Call:
## lm(formula = University.Rating ~ Serial.No. + GRE.Score + TOEFL.Score +
## SOP + LOR + CGPA + Research, data = test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.34352 -0.46556 -0.03557 0.44046 2.44809
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.4170319 1.2125399 -5.292 1.82e-07 ***
## Serial.No. 0.0001812 0.0002260 0.802 0.42307
## GRE.Score 0.0065910 0.0059476 1.108 0.26833
## TOEFL.Score 0.0209679 0.0103520 2.025 0.04336 *
## SOP 0.4474027 0.0507642 8.813 < 2e-16 ***
## LOR 0.1498125 0.0488318 3.068 0.00227 **
## CGPA 0.3578395 0.1141124 3.136 0.00182 **
## Research 0.0923371 0.0783086 1.179 0.23891
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7109 on 492 degrees of freedom
## Multiple R-squared: 0.619, Adjusted R-squared: 0.6135
## F-statistic: 114.2 on 7 and 492 DF, p-value: < 2.2e-16#Remove Serial Number
linear <- lm(University.Rating~ GRE.Score + TOEFL.Score + SOP +LOR + CGPA + Research, data = test )
summary(linear)##
## Call:
## lm(formula = University.Rating ~ GRE.Score + TOEFL.Score + SOP +
## LOR + CGPA + Research, data = test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.36251 -0.47140 -0.04223 0.45376 2.41297
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.295220 1.202548 -5.235 2.45e-07 ***
## GRE.Score 0.006468 0.005943 1.088 0.27705
## TOEFL.Score 0.020128 0.010295 1.955 0.05114 .
## SOP 0.441757 0.050255 8.790 < 2e-16 ***
## LOR 0.154072 0.048524 3.175 0.00159 **
## CGPA 0.364222 0.113793 3.201 0.00146 **
## Research 0.096184 0.078133 1.231 0.21890
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7106 on 493 degrees of freedom
## Multiple R-squared: 0.6185, Adjusted R-squared: 0.6138
## F-statistic: 133.2 on 6 and 493 DF, p-value: < 2.2e-16#Remove GRE
linear <- lm(University.Rating~ TOEFL.Score + SOP +LOR + CGPA + Research, data = test )
summary(linear)##
## Call:
## lm(formula = University.Rating ~ TOEFL.Score + SOP + LOR + CGPA +
## Research, data = test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.37560 -0.47448 -0.03629 0.45065 2.41676
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.243653 0.715856 -7.325 9.79e-13 ***
## TOEFL.Score 0.025353 0.009109 2.783 0.00559 **
## SOP 0.440906 0.050259 8.773 < 2e-16 ***
## LOR 0.151540 0.048478 3.126 0.00188 **
## CGPA 0.414718 0.103920 3.991 7.59e-05 ***
## Research 0.120784 0.074805 1.615 0.10702
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7107 on 494 degrees of freedom
## Multiple R-squared: 0.6176, Adjusted R-squared: 0.6137
## F-statistic: 159.5 on 5 and 494 DF, p-value: < 2.2e-16#Remove Research
linear <- lm(University.Rating~ TOEFL.Score + SOP +LOR + CGPA, data = test )
summary(linear)##
## Call:
## lm(formula = University.Rating ~ TOEFL.Score + SOP + LOR + CGPA,
## data = test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.46231 -0.46269 -0.04935 0.45262 2.39211
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.62010 0.67792 -8.290 1.07e-15 ***
## TOEFL.Score 0.02695 0.00907 2.971 0.00311 **
## SOP 0.44423 0.05030 8.832 < 2e-16 ***
## LOR 0.15563 0.04849 3.210 0.00142 **
## CGPA 0.44360 0.10254 4.326 1.83e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7119 on 495 degrees of freedom
## Multiple R-squared: 0.6155, Adjusted R-squared: 0.6124
## F-statistic: 198.1 on 4 and 495 DF, p-value: < 2.2e-16LogMod Backwards Selection Research
logmod <- glm(Research~., data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ ., data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.01724 -0.33223 0.00753 0.29143 0.99776
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.738e+00 7.931e-01 -5.974 4.44e-09 ***
## Serial.No. 7.426e-05 1.313e-04 0.565 0.572113
## GRE.Score 1.921e-02 3.327e-03 5.776 1.36e-08 ***
## TOEFL.Score -8.741e-03 5.980e-03 -1.462 0.144417
## University.Rating 2.412e-02 2.566e-02 0.940 0.347657
## SOP 1.441e-02 3.108e-02 0.464 0.643147
## LOR 1.404e-02 2.840e-02 0.494 0.621210
## CGPA -9.398e-02 7.457e-02 -1.260 0.208213
## Chance.of.Admit 1.065e+00 3.066e-01 3.474 0.000557 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1633439)
##
## Null deviance: 123.200 on 499 degrees of freedom
## Residual deviance: 80.202 on 491 degrees of freedom
## AIC: 523.91
##
## Number of Fisher Scoring iterations: 2plot(logmod)
#Removed LOR
logmod <- glm(Research~Serial.No. + GRE.Score + TOEFL.Score + University.Rating + SOP + CGPA, data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ Serial.No. + GRE.Score + TOEFL.Score +
## University.Rating + SOP + CGPA, data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.09472 -0.33171 0.01616 0.28395 1.02222
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.3439527 0.6572538 -9.652 < 2e-16 ***
## Serial.No. 0.0001857 0.0001291 1.438 0.151
## GRE.Score 0.0216466 0.0032778 6.604 1.04e-10 ***
## TOEFL.Score -0.0054357 0.0059720 -0.910 0.363
## University.Rating 0.0344431 0.0256319 1.344 0.180
## SOP 0.0302826 0.0298521 1.014 0.311
## CGPA 0.0443213 0.0648898 0.683 0.495
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1670782)
##
## Null deviance: 123.20 on 499 degrees of freedom
## Residual deviance: 82.37 on 493 degrees of freedom
## AIC: 533.24
##
## Number of Fisher Scoring iterations: 2#plot(logmod)
#Removed LOR, CGPA
logmod <- glm(Research~Serial.No. + GRE.Score + TOEFL.Score + University.Rating + SOP , data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ Serial.No. + GRE.Score + TOEFL.Score +
## University.Rating + SOP, data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.09070 -0.33673 0.01374 0.28546 1.03378
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.4208255 0.6471960 -9.921 < 2e-16 ***
## Serial.No. 0.0001936 0.0001285 1.506 0.133
## GRE.Score 0.0225907 0.0029705 7.605 1.45e-13 ***
## TOEFL.Score -0.0042408 0.0057070 -0.743 0.458
## University.Rating 0.0375039 0.0252235 1.487 0.138
## SOP 0.0358048 0.0287209 1.247 0.213
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1668978)
##
## Null deviance: 123.200 on 499 degrees of freedom
## Residual deviance: 82.448 on 494 degrees of freedom
## AIC: 531.72
##
## Number of Fisher Scoring iterations: 2#Removed LOR, CGPA, TOEFL
logmod <- glm(Research~Serial.No. + GRE.Score + University.Rating + SOP , data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ Serial.No. + GRE.Score + University.Rating +
## SOP, data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.09592 -0.34393 0.00147 0.29124 1.03427
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.3837643 0.6449795 -9.898 <2e-16 ***
## Serial.No. 0.0002017 0.0001280 1.575 0.116
## GRE.Score 0.0210992 0.0021887 9.640 <2e-16 ***
## University.Rating 0.0346801 0.0249243 1.391 0.165
## SOP 0.0319968 0.0282472 1.133 0.258
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1667468)
##
## Null deviance: 123.20 on 499 degrees of freedom
## Residual deviance: 82.54 on 495 degrees of freedom
## AIC: 530.27
##
## Number of Fisher Scoring iterations: 2#Removed LOR, CGPA, TOEFL, SOP
logmod <- glm(Research~Serial.No. + GRE.Score + University.Rating , data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ Serial.No. + GRE.Score + University.Rating,
## data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.10835 -0.34957 0.00049 0.28952 1.02269
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.5389338 0.6304444 -10.372 <2e-16 ***
## Serial.No. 0.0001855 0.0001272 1.458 0.1455
## GRE.Score 0.0217887 0.0021030 10.361 <2e-16 ***
## University.Rating 0.0504027 0.0207077 2.434 0.0153 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.166842)
##
## Null deviance: 123.200 on 499 degrees of freedom
## Residual deviance: 82.754 on 496 degrees of freedom
## AIC: 529.57
##
## Number of Fisher Scoring iterations: 2#Removed LOR, CGPA, TOEFL, SOP, Serial Number
logmod <- glm(Research~ GRE.Score + University.Rating , data=test)
summary(logmod)##
## Call:
## glm(formula = Research ~ GRE.Score + University.Rating, data = test)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.14033 -0.35017 0.00906 0.29255 1.00181
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.415603 0.625451 -10.258 <2e-16 ***
## GRE.Score 0.021546 0.002099 10.266 <2e-16 ***
## University.Rating 0.050337 0.020731 2.428 0.0155 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.1672199)
##
## Null deviance: 123.200 on 499 degrees of freedom
## Residual deviance: 83.108 on 497 degrees of freedom
## AIC: 529.71
##
## Number of Fisher Scoring iterations: 2plot(logmod)
#based on the logMod summary, the 2 most signifant variables are University Rating and GRE.Score. Based on these results, as the University Rating increases, so does then number of students who do conduct Research.
CVs
CV for linear model - Chance of Admission with CGPA - Manual Leave on Out
set.seed(7861)
cvlm <- list()
msecv <- NA
for(i in 1:nrow(test)){
#Fit the linear model
cvlm[[i]] <- lm(Chance.of.Admit[-i] ~ CGPA[-i])
# Calculate MSE for ith model
msecv[i] <- (predict(cvlm[[i]], newdata = data.frame(CGPA[-i]))-Chance.of.Admit[i])^2
#msecv[i]
}
#output mean of MSE
mean(msecv)## [1] 0.06879746CV for linear model - Chance of Admittion - Manual Leave on Out
set.seed(7861)
cvlm <- list()
msecv <- NA
for(i in 1:nrow(test)){
#Fit the linear model
cvlm[[i]] <- lm(Chance.of.Admit[-i] ~ Serial.No.[-i] + GRE.Score[-i] + TOEFL.Score[-i] +LOR[-i] + CGPA[-i] + Research[-i])
# Calculate MSE for ith model
msecv[i] <- (predict(cvlm[[i]], newdata = data.frame(Serial.No.[-i] + GRE.Score[-i] + TOEFL.Score[-i] +LOR[-i] + CGPA[-i] + Research[-i]))-Chance.of.Admit[i])^2
#msecv[i]
}
#output mean of MSE
mean(msecv)## [1] 0.0666215CV for linear model - Research
ResearchData <- test$Research
ResearchDataFactor <- factor(test$Research)
simlog<-glm(factor(Research)~., family = "binomial", data = test)
table(predict(simlog, type = "response")>0.5, ResearchData)## ResearchData
## 0 1
## FALSE 154 57
## TRUE 66 223misclassificationRate <- (57+66)/(154+223)
capture.output(cat('Misclassification rate = ', misclassificationRate))## [1] "Misclassification rate = 0.3262599"library(MLmetrics)##
## Attaching package: 'MLmetrics'## The following object is masked from 'package:base':
##
## RecallF1<- F1_Score(as.numeric(predict(simlog, type = "response")>0.5), ResearchData)
Accu <- Accuracy(as.numeric(predict(simlog, type = "response")>0.5), ResearchData)
Sens <- Sensitivity(as.numeric(predict(simlog, type = "response")>0.5), ResearchData)
scoreTable <-cbind(F1, Accu, Sens)
colnames(scoreTable)<-c("F1 Score", "Accuracy", "Sensitivity")
rownames(scoreTable)<-c("Logistic Regression")
#rownames(scoreTable)<-c("Logistic Regression", "Neural Network")
round(scoreTable,3)## F1 Score Accuracy Sensitivity
## Logistic Regression 0.715 0.754 0.73CV for linear model - University Rating - Manual Leave on Out
set.seed(7861)
cvlm <- list()
msecv <- NA
for(i in 1:nrow(test)){
#Fit the linear model
cvlm[[i]] <- lm(University.Rating[-i] ~ TOEFL.Score[-i] + SOP[-i] + LOR[-1] + CGPA[-i])
# Calculate MSE for ith model
msecv[i] <- (predict(cvlm[[i]], newdata = data.frame(TOEFL.Score[-i] + SOP[-i] + LOR[-1] + CGPA[-i]))-University.Rating[i])^2
#msecv[i]
}
#output mean of MSE
mean(msecv)## [1] 3.373402